Spurious Quasi-Resonances in Boundary Integral Equations for the Helmholtz Transmission Problem

We consider the Helmholtz transmission problem with piecewise-constant material coefficients, and the standard associated direct boundary integral equations. For certain coefficients and geometries, the norms of the inverses of the boundary integral operators grow rapidly through an increasing sequence of frequencies, even though this is not the case for the solution operator of the transmission problem; we call this phenomenon that of spurious quasi-resonances. We give a rigorous explanation of why and when spurious quasi-resonances occur, and propose modified boundary integral equations that are not affected by them.



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1 Introduction and summary of the main results

1.1 The Helmholtz transmission scattering problem

We consider the scattering of an incident time-harmonic acoustic wave by a penetrable homogeneous object that occupies the region of space , , which is a bounded Lipschitz open set. We first introduce notation necessary for a precise mathematical statement of this transmission problem. Let , , and let

be the unit normal vector field on

pointing from into . For any , we let and . With , we define the Dirichlet and Neumann trace operators

with and such that if then . Let be the Cauchy trace, which satisfies

Given , for some ball , and , satisfies the Sommerfeld radiation condition if


uniformly in all directions, where ; we then write .

Given and frequency , the Helmholtz transmission scattering problem is that of finding the complex amplitude of the sound pressure, with the solution of


where the incident wave is an entire solution of the homogeneous Helmholtz equation in ,


This set up means that is the total field in and the scattered field in .

In principle, the jump of the Cauchy trace of across can be more general than the Cauchy trace of an incident wave. This leads to the following generic Helmholtz transmission problem.

Definition 1.1.

(The Helmholtz transmission problem.) Given positive real numbers and and , find such that,


The following well-posedness result is proved in, e.g., [23, Lemma 2.2 and Appendix A].

Lemma 1.2.

The solution of the transmission problem of Definition 1.1 exists and is unique. Moreover, if then .

Remark 1.3.

The transmission problem of Definition 1.1 is not the most general form of the transmission problem. If the transmission condition in (1.4) is replaced by


for a constant, then this covers all possible constant-coefficient transmission problems; see, e.g., [23, Page 322]. In Appendix A we outline how our results extend this more general transmission problem. We see that, although the main ideas remain the same, more notation and technicalities are required, hence why we have chosen to focus on the simpler problem of Definition 1.1.

1.2 Solution operators and quasi-resonances

Definition 1.4.

(Solution operators.) Given positive real numbers and , let

where is the solution of the Helmholtz transmission problem


Lemma 1.2 implies that is well defined and bounded on either or . We introduce the abbreviations

We refer to as the “physical” solution operator, since it corresponds to the transmission problem of Definition 1.1, and as the “unphysical” solution operator, since it corresponds to the transmission problem where the indices and are swapped compared to those in Definition 1.1.

Recall that the high-frequency behaviour of depends on which of and is larger. Indeed, if and is Lipschitz and star-shaped with respect to a ball, then Lemma 4.5 below shows that the norm of has, at worst, mild algebraic growth in ; this result uses the bounds on the solution operator from [23], with analogous bounds obtained for smooth, convex with strictly positive curvature in [9]. If and is smooth and convex with strictly positive curvature, then Lemma 4.6 below, based on the results of [28], shows that there exists with such that the norm of blows up superalgebraically through as . (Similar results in the particular case when is a ball were obtained in [7, 8], and summarised in [1, Chapter 5]).

We call these real frequencies quasi-resonances, since they can be understood as real parts of complex resonances of the transmission problem lying close to the real axis (with this terminology also used in, e.g., [7, 8, 1]); the particular functions on which at blows up are called quasimodes. The relationship between quasimodes and resonances is a classic topic in scattering theory; see [33, 31, 32], [17, §7.3]. The Weyl-type bound on the number of resonances of the transmission problem when is smooth and convex with strictly positive curvature in [10, Theorem 1.3] implies that the number of quasi-resonances in in this case grows like as .

Remark 1.5.

The physical reason for the existence of quasi-resonances when is that, in this case, geometric-optic rays can undergo total internal reflection when hitting from . Rays “hugging” the boundary via a large number of bounces with total internal reflection correspond to solutions of the transmission problem localised near

; in the asymptotic-analysis literature these solutions are known as “whispering gallery” modes; see, e.g.,

[3, 4]. The existence of quasi-resonances of the transmission problem has only been rigorously established when is smooth and convex with strictly positive curvature. The understanding above via rays suggests that such quasi-resonances and quasimodes do not exist for polyhedral (since sharp corners prevent rays from moving parallel to the boundary), although solutions with localisation qualitatively similar to that of quasimodes can be seen when is a pentagon [20, Figure 13] or a hexagon [6, Figure 23].

1.3 Calderón projectors and the standard first- and second-kind direct boundary integral equations (BIEs)

Since all the layer potentials and integral operators depend on , we omit this -dependence in the notation. Let the Helmholtz fundamental solutions be given by

where is the Hankel function of the first kind and order zero; see [29, Section 3.1].

As in [29, Equation 3.6], the single-layer, adjoint-double-layer, double-layer, and hypersingular operators are defined for and by


for (note that the sign of the hypersingular operator is swapped compared to, e.g., [11]). When is Lipschitz, the integrals defining and must be understood as Cauchy principal values (see, e.g., [11, Equation 2.33]), and the integral defining is understood as a non-tangential limit (see, e.g., [11, Equation 2.36]) or finite-part integral (see, e.g., [21, Theorem 7.4 (iii)]), but we do not need the details of these definitions in this paper.

Let the Calderón projectors be defined by


see, e.g., [29, Section 3.6], [11, Page 117]. Basic results about (including that they are indeed projectors) are in §2, but we record here that


Let the boundary integral operators (BIOs) and be defined by



Lemma 1.6.

If is the solution of the Helmholtz transmission problem of Definition 1.1, then


In particular, if solves the Helmholtz transmission scattering problem (1.2), then


These boundary integral equations (BIEs) are called single-trace formulations (STFs). The first-kind BIEs in (1.13) and (1.14) appeared in [16], [37], and are also derived in, e.g., [13, Section 3.3]. Their counterparts for electromagnetic scattering are known as the PMCHWT (Poggio–Miller–Chang–Harrington–Wu–Tsai) formulation [27]. The second-kind BIEs in (1.13) and (1.14) can be found in, e.g., [14] and are known as the Müller formulation in computational electromagnetics [25].

Lemma 1.7.

(i) Both and are bounded and invertible on .

(ii) is bounded and invertible on .

The proofs of Lemmas 1.6 and 1.7 are contained in §2.

1.4 Spurious quasi-resonances for the standard BIOs

Lemma 1.7 shows that the BIEs of (1.13) and (1.14) are well-posed. It is then reasonable to believe that the solution operators of these BIEs inherit the behaviour (with respect to frequency) of the solution operator of the transmission problem. The following numerical results, however, show that this is not the case. 111The code used to produce the numerical results is available at https://github.com/moiola/TransmissionBIE-OpNorms

Example 1.8.

If is a circle for or a sphere for all boundary integral operators , , , and can be “diagonalized” by switching to a “modal”

-orthogonal basis of Fourier harmonics in 2D or spherical harmonics in 3D, respectively. The corresponding eigenvalues can be found in, e.g.,

[2] for and in, e.g., [36] for . All relevant norms have a simple sum representation with respect to these bases. Therefore we can compute the norms of the solution operators as the maximum of the Euclidean norms of -matrices, one for every mode. We did this in MATLAB for the modes of order at most 100, which seems to be sufficient, because the maximal norm was invariably found among the modes of order .

We report the computed norms of the solution operator along with the norms of and (i.e., the solution operators for the BIEs (1.14)) on the space , where we use the weighted norm defined in §4.1. We plot these norms for different frequencies and give the results for in Figure 1 and for in Figure 2.

Figure 1: unit circle: Norms of operators , , and on for , (left) and , (right)
Figure 2: unit sphere: Norms of operators , , and on for , (left) and , (right)

When (plots on the left) we see the typical spikes in the norms as a function of , expected because of the results recalled in §1.2; these spikes are caused by quasi-resonances. Conversely, for (right plots) the norm of (in yellow) does not have any spikes, whereas the spikes persist in the norms of and .

The observations made in Example 1.8 provide evidence of spurious quasi-resonances of and when : for certain frequencies these boundary integral operators are ill-conditioned though for the same frequencies the solution operator is stable.

On rare occasions such spurious quasi-resonances have been noticed before. Indeed, the paper [22] computed the complex eigenvalues of and and pointed out in [22, Section 2.3] the existence of “fictitious eigenvalues”, i.e., non-physical poles of the resolvent operators. Although [22] did not give a rigorous explanation for this phenomenon, [22] attempted to remedy it by modifying the BIEs; these new BIEs, however, still have issues with poles with small imaginary part – see the discussion in [22, §4]. Non-physical spikes in the condition numbers of discretized BIEs for Helmholtz transmission problems were also reported in [34, Section 4.4], but no deeper investigation was attempted.

The observation of the spurious quasi-resonances of Example 1.8 was the starting point for this paper – we wanted to understand precisely why they affect and . We also wanted to find alternative BIEs immune to spurious quasi-resonances. The remainder of this paper reports our progress towards these goals.

Remark 1.9.

For the standard first and second-kind BIEs for the exterior Dirichlet and Neumann problems for the Helmholtz operator (modelling acoustic scattering by impenetrable objects), the occurrence of spurious (true) resonances is well-known; see, e.g., [29, Section 3.9.2]: the solutions of the BIEs are not unique for an infinite sequence of distinct s, although the boundary-value problems have unique solutions for all . The standard remedies for this are recalled (and linked to the results of the present paper) in Remark 1.16 below.

1.5 The main results

1.5.1 The relationship between the BIOs and the solution operators

Theorem 1.10.

As an operator on , has the decomposition


and, as an operator on either or , has the decomposition


The following result uses (1.15) and results about the behaviour of and in Lemmas 4.5 and 4.6 below to prove that if , then the norm of blows up through the quasi-resonances of the transmission problem (1.6) with and . This result explains rigorously the experiments in Figures 1 and 2. The result is stated using the weighted norm defined in §4.1, with the operator norm

Theorem 1.11.

(Superalgebraic blow up of for smooth and convex.) If is with strictly-positive curvature and , then there exist frequencies with such that given any there exists such that

The reason we only prove blow up of , and not of , is that Theorem 1.10 shows that involves not only and but also the composition of and (whereas does not), and we do not currently know how to show that this extra term does not cancel out the blow up of one of or .

The next result shows that, on appropriate subspaces, and involve only the physical solution operator . In particular, this result demonstrates that, because of the specific form of the right-hand sides in (1.13), only the physical solution operator is involved in the solution of the boundary value problem of Definition 1.1, as expected. The results for hold on either or , but the results for hold only on (since we have not proved that exists on ). We use the notation that is the range of the operator .

Theorem 1.12.

( and as operators .)

(i) .

(ii) Both and are bounded and invertible from with as operators from .

1.5.2 Augmented BIEs

We now propose a simple way to suppress spurious quasi-resonances in the BIEs without resorting to products of integral operators. We work in the Hilbert space where for the results involving , and equals either or for the results involving ; the norm is then either or . We equip the space with the norm

where with .

Define the augmented BIOs and by


The idea behind introducing these augmented systems is that the solution to the BIEs (1.13) satisfies (we see this below in (2.10) in the proof of Lemma 1.6).

Lemma 1.13.

(Solutions of augmented BIEs.) Let be one of and . Given , if the solution to the augmented system


exists, then satisfies


and is given by


Lemma 1.13 shows that the solution of the augmented system (1.19), if it exists, only involves the physical solution operator . Note that if , i.e., the right-hand side of the first- and second-kind BIEs (1.13), then (1.20) is satisfied; indeed, it follows from Lemma 3.2 below that .

Example 1.14.

As in Example 1.8 we perform a “diagonalization” of the augmented BIOs of (1.18) to compute the operator norms of their pseudo-inverses in

numerically (i.e., we compute the inverse of the smallest singular value of the block-diagonal matrix arising from truncating the Fourier/spherical-harmonic expansion). These norms as functions of the frequency

are plotted in Figure 3 for the case , , in which the physical solution operator has small norm for all values of considered (as shown by the right-hand plots of Figures 1 and 2).

Figure 3: Plots of the operator norms of the pseudo-inverses , of the augmented BIOs

As an agreeable surprise, we see that the norms of the pseudo-inverses of the augmented BIOs are smaller than those of for the range of frequencies considered – augmentation has successfully removed any spurious quasi-resonances!

The following theorem rigorously explains the results in Figure 3.

Theorem 1.15.

(Stability of augmented BIEs.)




This theorem reveals that the operator norms of the pseudo-inverses and are bounded by for some -independent constant . Hence, if the physical solution operator is well-conditioned, then this well-conditioning carries over to the BIOs of the augmented formulations.

Remark 1.16.

(The analogue of Theorem 1.10 for BIOs for scattering by impenetrable obstacles.) The analogous formulae to those in Theorem 1.10 for second-kind combined-field BIOs for solving the exterior Dirichlet, Neumann, and impedance problems were given in [11, Theorem 2.33], with formulae for certain BIOs involving operator preconditioning given in [5, Lemma 6.1]. (We note that [5, Lemma 6.1] introduced the idea of obtaining these formulae via Calderón projectors, and we prove Theorem 1.10 using this idea in §3.)

For example, the standard direct second-kind combined-field BIO for solving the exterior Dirichlet problem involves the operator , for , and [11, Theorem 2.33] and [5, Lemma 6.1] prove that


where is the exterior Dirichlet-to-Neumann map for solutions of the Helmholtz equation satisfying the Sommerfeld radiation condition (1.1), and is the interior impedance-to-Dirichlet map (where the impedance boundary condition is ). Recalling that is also the standard indirect second-kind BIO for solving the interior impedance problem, we see that (1.24) expresses in terms of the solution operators for the appropriate exterior and interior problems solved using .

The standard indirect second-kind combined-field BIO for solving the exterior Dirichlet problem involves the operator ; this operator is also the standard direct second-kind BIO for solving the interior impedance problem, and, correspondingly,

Remark 1.17.

(Indirect BIEs) In this paper, we have considered only direct BIEs for the Helmholtz transmission problem, i.e., BIEs where the unknown is the Cauchy data of the solution. It is reasonable to expect that similar results hold for indirect BIEs for the transmission problem, just as similar decompositions into solution operators hold for the inverses of the direct BIOs for scattering by impenetrable obstacles (see the previous remark and [11, Theorem 2.33]), but we have not investigated this.

Remark 1.18.

(Spurious quasi-resonances for electromagnetic BIEs) We expect that the phenomenon of spurious quasi-resonances also occurs for the BIEs for time-harmonic electromagnetic scattering; we have not pursued this in this paper however.

2 Recap of results about layer potentials, BIOs, and Calderón projectors

The single-layer and double-layer potentials, and respectively, are defined for by


these definitions for naturally extend to for by continuity (see, e.g., [11, Page 109]).

Lemma 2.1.

(i) If with , then .

(i) If with , then .

References for the proof.

See, e.g., [11, Theorem 2.15]; we note that the mapping properties for crucially use the harmonic analysis results of [15], [35]. ∎

The potentials (2.1) and (2.2) are related to the integral operators in (1.7) and (1.8) via the jump relations


see, e.g., [21, §7, Page 219]. Recall the mapping properties, valid when is Lipschitz, , and ,


see, e.g., [11, Theorems 2.17 and 2.18] (similar to the results of Lemma 2.1, the mapping properties for crucially use the harmonic analysis results of [15], [35]). The mapping properties (2.4) imply that is a bounded operator from to itself and from to itself.

We use the following notation for spaces of Helmholtz solutions:

Lemma 2.2.



By the jump relations (2.3) and the definitions of (1.9), with ,


see, e.g., [11, Equation 2.49]. Both when and when , the right-hand side is then the trace of an element of by Lemma 2.1, so that . To prove the reverse inclusion, given , by Green’s integral representation (see, e.g., [11, Theorems 2.20 and 2.21]); (2.5) with and then implies that . ∎

The following two lemmas are proved in, e.g., [11, Page 118 and Lemma 2.22], respectively.222 Strictly speaking, [11, §2.5] only considers as operators on , but the proofs of the results on are the same.

Lemma 2.3.

and as operators either on or on .

Lemma 2.4.

(i) If then


(ii) If then


The next lemma is a converse to Lemma 2.4.

Lemma 2.5.

Let or .

(i) If , then for some .

(ii) If , then for some .


(i) Given such that , let


By Lemma 2.1, . We therefore only need to check that . However, by (2.5) and the definition of (2.8), .

(ii) Given such that , let


Similar to in (i), , and, by (2.5) and the definition of (2.9),

Proof of Lemma 1.6.

By (1.4) and (2.6), , so that, by (1.10),


Similarly, by (1.4), (2.7), and (1.10),


Applying to the transmission condition in (1.4) and using (2.11), we find that


Subtracting (2.10) from (2.12), we obtain the first-kind BIE in (1.13). Adding (2.10) to (2.12), we obtain the second-kind BIE in (1.13). ∎

Remark 2.6.

If with satisfying (1.3), then by (2.6), and thus the right-hand sides of (1.13) are just .

Lemma 2.7.

Let equal either or . Then is an injective, bounded operator on either or .


The boundedness of follows from the expressions (1.11)/(1.12) and the boundedness of . Injectivity follows by repeating the arguments in the proof of Theorem 1.10 below with (these arguments use uniqueness of the Helmholtz transmission problem of Definition 1.1). ∎

Proof of Lemma 1.7.

The result for follows from Lemma 2.7 combined with the coercivity result of, e.g., [13, Theorem 7.27] (see [13, Corollary 7.28]).

The results for follows from Lemma 2.7 combined with the fact that is compact on both